13. Three-Dimensional Coordinate Systems:
- I hope you guys know this by heart already.
.. there's a distance formula, and not much else.
- This section is also pretty basic, and lays out vector addition and scalar multiplication. You guys
should know this pretty well too.
13.3 The Dot Product:
- For the underinformed, the dot product gives a SCALAR value, not a vector value. i.e.
<(1,2,3),(3,2,1)> = (1)(3)+(2)(2)+(3)(1)
- The formula for the angle between two vectors, cos(theta) = a.b/(|a||b|), is useful, both in
computing a.b and the angle theta.
- The formulas for scalar and vector projections are probably less fundamental (in my opinion), but
as always, if you have time, it can be useful to know these formulas.
13.4 The Cross Product
- The cross product of two vectors, axb, gives a vector orthogonal to both a and b via the right hand
rule. This is, in my opinion, trickier than the dot product, and it is important to briefly remind
yourself about the basic properties (i.e. bxa = -axb, etc.)
- (trick): Two vectors a, b are parallel iff axb = 0. Sometimes it is easier to just compute the cross
product than to try to find lambda with a = lambda*b.
- The scalar triple product ((axb).c) = det([a b c]) and the formula sin(theta) = |axb|/(|a||b|) are good
to know, but probably less fundamental than other cross product facts, and the sin(theta) formula is
not used as much as the cos(theta) formula.
13.5 Equations of Lines and Planes
- A point and a normal vector uniquely determine a plane;
- Three (non-colinear) points uniquely determine a plane;
- A line and a point (not lying on that line) uniquely determine a plane;
- An important form for the equation of a plane is <v,((x,y,z)-p)> = 0, where p is any point on the
plane, and v is the normal vector.
- There are two forms for an equation of a line: the parametric equations (more useful, I think), and
the symmetric equations.
13.6 Cylinders and Quadric Surfaces
- The table on p.872 is pretty useful, and worth memorizing. Break down the quadrics into ones
with constants in the equation (ellipsoid, hyperboloids of (one and two) sheets, cylinder), and
without (cone, (elliptic and hyperbolic) paraboloids).
- As an example of a possible mnemonic to remember the surfaces further, all of the non-paraboloid
surfaces have all quadratic terms (x^2,y^2,z^2).
- The traces (most of the time) will give you the name of the surface.